IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v121y2011i10p2231-2242.html
   My bibliography  Save this article

Convergence to type I distribution of the extremes of sequences defined by random difference equation

Author

Listed:
  • Hitczenko, Pawel

Abstract

We study the extremes of a sequence of random variables (Rn) defined by the recurrence Rn=MnRn-1+q, n>=1, where R0 is arbitrary, (Mn) are iid copies of a non-degenerate random variable M, 0 0 is a constant. We show that under mild and natural conditions on M the suitably normalized extremes of (Rn) converge in distribution to a double-exponential random variable. This partially complements a result of de Haan, Resnick, Rootzén, and de Vries who considered extremes of the sequence (Rn) under the assumption that .

Suggested Citation

  • Hitczenko, Pawel, 2011. "Convergence to type I distribution of the extremes of sequences defined by random difference equation," Stochastic Processes and their Applications, Elsevier, vol. 121(10), pages 2231-2242, October.
    • Handle: RePEc:eee:spapps:v:121:y:2011:i:10:p:2231-2242
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414911001438
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Devroye, Luc & Fawzi, Omar, 2010. "Simulating the Dickman distribution," Statistics & Probability Letters, Elsevier, vol. 80(3-4), pages 242-247, February.
    2. de Haan, Laurens & Resnick, Sidney I. & Rootzén, Holger & de Vries, Casper G., 1989. "Extremal behaviour of solutions to a stochastic difference equation with applications to arch processes," Stochastic Processes and their Applications, Elsevier, vol. 32(2), pages 213-224, August.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Marco Rocco, 2011. "Extreme value theory for finance: a survey," Questioni di Economia e Finanza (Occasional Papers) 99, Bank of Italy, Economic Research and International Relations Area.
    2. Aoki, Masanao, 2002. "Open models of share markets with two dominant types of participants," Journal of Economic Behavior & Organization, Elsevier, vol. 49(2), pages 199-216, October.
    3. Danielsson, Jon & Jorgensen, Bjorn N. & Sarma, Mandira & de Vries, Casper G., 2006. "Comparing downside risk measures for heavy tailed distributions," Economics Letters, Elsevier, vol. 92(2), pages 202-208, August.
    4. Phornchanok Cumperayot & Casper G. de Vries, 2006. "Large Swings in Currencies driven by Fundamentals," Tinbergen Institute Discussion Papers 06-086/2, Tinbergen Institute.
    5. Cotter, John, 2007. "Varying the VaR for unconditional and conditional environments," Journal of International Money and Finance, Elsevier, vol. 26(8), pages 1338-1354, December.
    6. Collamore, Jeffrey F. & Vidyashankar, Anand N., 2013. "Tail estimates for stochastic fixed point equations via nonlinear renewal theory," Stochastic Processes and their Applications, Elsevier, vol. 123(9), pages 3378-3429.
    7. Gomes, M. Ivette & Hall, Andreia & Miranda, M. Cristina, 2008. "Subsampling techniques and the Jackknife methodology in the estimation of the extremal index," Computational Statistics & Data Analysis, Elsevier, vol. 52(4), pages 2022-2041, January.
    8. Chi, Zhiyi, 2016. "On exact sampling of the first passage event of a Lévy process with infinite Lévy measure and bounded variation," Stochastic Processes and their Applications, Elsevier, vol. 126(4), pages 1124-1144.
    9. Jondeau, Eric & Rockinger, Michael, 2003. "Testing for differences in the tails of stock-market returns," Journal of Empirical Finance, Elsevier, vol. 10(5), pages 559-581, December.
    10. Gimeno, Ricardo & Gonzalez, Clara I., 2012. "An automatic procedure for the estimation of the tail index," MPRA Paper 37023, University Library of Munich, Germany.
    11. Danielsson, Jon & Ergun, Lerby M. & Haan, Laurens de & Vries, Casper G. de, 2016. "Tail index estimation: quantile driven threshold selection," LSE Research Online Documents on Economics 66193, London School of Economics and Political Science, LSE Library.
    12. Lux, Thomas & Alfarano, Simone, 2016. "Financial power laws: Empirical evidence, models, and mechanisms," Chaos, Solitons & Fractals, Elsevier, vol. 88(C), pages 3-18.
    13. Weissman, Ishay, 2023. "Some comments on “The density flatness phenomenon” by Alhakim and Molchanov and the Dickman distribution," Statistics & Probability Letters, Elsevier, vol. 194(C).
    14. P. Hartmann & S. Straetmans & C. G. de Vries, 2004. "Asset Market Linkages in Crisis Periods," The Review of Economics and Statistics, MIT Press, vol. 86(1), pages 313-326, February.
    15. Kevin Dowd & John Cotter, 2011. "Intra-Day Seasonality in Foreign Market Transactions," Working Papers 200746, Geary Institute, University College Dublin.
    16. François, LONGIN & Bruno, SOLNIK, 1998. "Correlation Structure of International Equity Markets During Extremely Volatile Periods," HEC Research Papers Series 646, HEC Paris.
    17. Wagner, Niklas & Marsh, Terry A., 2005. "Measuring tail thickness under GARCH and an application to extreme exchange rate changes," Journal of Empirical Finance, Elsevier, vol. 12(1), pages 165-185, January.
    18. Groenendijk, Patrick A. & Lucas, Andre & de Vries, Casper G., 1995. "A note on the relationship between GARCH and symmetric stable processes," Journal of Empirical Finance, Elsevier, vol. 2(3), pages 253-264, September.
    19. Liu, Ji-Chun, 2006. "On the tail behaviors of Box-Cox transformed threshold GARCH(1,1) process," Statistics & Probability Letters, Elsevier, vol. 76(13), pages 1323-1330, July.
    20. Chaudhuri, Kausik & Sen, Rituparna & Tan, Zheng, 2018. "Testing extreme dependence in financial time series," Economic Modelling, Elsevier, vol. 73(C), pages 378-394.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:121:y:2011:i:10:p:2231-2242. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.